Fundamental single-qubit quantum gates called phase shift gates change the phase of a qubit’s quantum state. Phase shift gates change the phase factor of the qubit, unlike bit-flip gates such as the Pauli-X or NOT gate which alter the qubit’s probability amplitudes. Another term for the phase shift gate is a phase gate.
phase shift gate
Functionality: A phase shift gate rotates the state vector of the qubit around the Bloch sphere’s z-axis. This modulates the relative phase between the qubit’s |0⟩ and |1⟩ components of superposition.
where φ is the angle of the phase shift.
Action on Basis States:
- When applied to the state |0⟩, the phase shift gate does not change the state: |0⟩ → |0⟩.
- When applied to the state |1⟩, the phase shift gate multiplies the state by a complex phase factor: |1⟩ → eiφ|1⟩.
Effect on Superposition: When applied to a qubit in a superposition state, such as α|0⟩ + β|1⟩, the gate transforms the state to α|0⟩ + eiφβ|1⟩. This changes the relative phase between the |0⟩ and |1⟩ components of the superposition.
Phase shift gate Matrix Representation
The general form of a phase shift gate can be represented by the following 2×2 unitary matrix:
where φ is the angle of rotation around the z-axis of the Bloch sphere.
Phase shift gate Types
Several important phase shift gates are used in quantum computation:
Z gate
The Z gate, or Pauli-Z gate, is a phase shift gate with φ = π. It flips the sign of the |1⟩ component:
This gate transforms |0⟩ to |0⟩ and |1⟩ to -|1⟩.
it means Z|0⟩=|0⟩, Z|1⟩=-|1⟩
S gate
Also known as the π/2-phase gate, or simply the phase gate, or the Z90 gate. the S gate has φ = π/2. It transforms |0⟩ to |0⟩ and |1⟩ to i|1⟩.
it means S|0⟩=|0⟩, S|1⟩=i|1⟩
T gate:
The T gate, sometimes called the π/8-gate, has φ = π/4. It transforms |0⟩ to |0⟩ and |1⟩ to eiπ/4|1⟩.
T|0⟩=|0⟩, T|1⟩=eiπ/4|1⟩
Phase shift and phase kickback:
- The phase kickback method uses a phase shift gate to move an operator’s eigenvalue from a target qubit to the control qubit via a CNOT gate.
- Application of a phase-query in quantum algorithms depends on the phase kickback method.
- Phase estimation techniques depend also on the capacity to translate data into qubit phases. The eigenvalues of a unitary operator can be found by use of the phase estimation method.
Universality: Any quantum computing may be built using phase shift gates, coupled with other single-qubit gates such as the Hadamard gate, and with multi-qubit gates such the CNOT gate. Since the T gate is not in the Clifford group of quantum gates as are the Z, S, and Hadamard gates, it is especially crucial to attain universality from this angle.
Implementation: Phase shift gates are physically manifested by qubit resonance frequency manipulation. For superconducting qubits, for instance, the period of an excitation at the qubit’s resonance frequency controls the phase rotation.
Visions
- Especially underlined as a building block for more complicated quantum gates and circuits is the S gate.
- Since it is a Clifford gate, fault-tolerant quantum computing finds application in it.Though it is not a Clifford gate, the T gate—also known as the π/8 gate—is very essential for reaching universality in quantum processing. Forming a universal set of quantum gates, it balances the Clifford gates.
- The use of phase shift gates in controlling the phase coherence of quantum states, which is essential for using interference-based quantum algorithms such as Grover’s search algorithm or Shor’s algorithm.
- The PhaseGate class in Qiskit offers a flexible solution for any phase changes, enabling developers to apply certain phases for bespoke circuit designs.
Phase shift gate Uses in quantum computing
- Phase shift gates are essential parts of quantum algorithms including those used in quantum computers:
- Phase shift gates are used to control the phases of qubits in the QFT, a fundamental subroutine in many quantum systems including Shor’s factoring method.
- Controlled phase shift gates help to estimate the phase of eigenvalues of unitary operators.
- Grover’s Search Algorithm amplifies the amplitude of the intended state using phase shift operations.
- Controlled-phase and multi-qubit gates are built from circuits designed for this purpose.
- Error Correction: Quantum computing supported by the Clifford gates—including the S gate—is fault-tolerant.